The remainder-form decomposition functions returned by Algorithm REF  can be used with the generalized embedding system in Definition REF 
to over-approximate (unconstrained) reachable sets of a dynamic system governed by the vector field \(f(\cdot )\) , which corresponds to the propagated/predicted sets in state observers/estimators. However, when additional state constraint information is available (e.g., sensor observations/measurements in state estimation problems, known safety constraints from system design and manufactured constraints from modeling redundancy [1]}, [2]}, [3]}, [4]}, [5]}), an additional set inversion (also known as update or refinement) step will allow us to take the advantage of the constraints to shrink the propagated sets, i.e., to obtain a tighter subset of the propagated set that is compatible/consistent with the given constraints. Further details about the application of the reachable/propagated set and set inversion algorithms will be described in Section .
